Absolute continuity in infinite dimensions and anticipating stochastic calculus
Let the Banach space of sequences \(B=R^\infty\) be endowed with the Borel \(\sigma\)-algebra and the probability \(P= \lambda^{\otimes \infty}\), where the measure \(\lambda\) is allowed to be either the standard Gaussian distribution of the real line \(R\), or the exponential law on the positive half line \(R^+\), or the uniform distribution over the interval \([-1,1]\). The problem of absolute continuity of a perturbation of the identity \(I_B+F\) in the Gaussian case has been studied by several authors, among them Girsanov (1960), Ramer (1970), Kuo (1975) and Kusuoka (1982), and more recently by Buckdahn, Üstünel/Zakai and others. The case of an exponential distribution \(\lambda\) which corresponds to an anticipative Girsanov theorem on the Poisson space has been studied recently by the author. Here, in this paper, the author extends the results known from the Gaussian case and from the exponential one to the case of the uniform distribution \(\lambda\) by developping a unified approach which is common for these three cases. So he presents in Section 2 a unified framework for the stochastic calculus of variations, the Sobolev spaces and the integration by parts formula, in order to study sufficient conditions for the absolute continuity of the transformation \(I_B+F: B\to B\) and to give an explicit expression of its density with divergence operator and Carleman-Fredholm determinant, first under an invertibility assumption, and then, in Section 5, under exclusion of the case of uniform distribution \(\lambda\), its generalization to non-invertible transformations. In this generalization in Section 5 the authors follows ideas of Üstünel/Zakai (1994) for the Gaussian case, and its own ideas from 1996 for the exponential case. The interesting paper is well written, and the unified approach with notations well known to the reader which is familiar with the framework in the Gaussian case should be appreciated by him.
scientific article; zbMATH DE number 1199901